On a result of Telyakovskij and multiple Hilbert transforms with polynomial phases. (Q1889530)

The author first mentions as Theorem 1, a result on an estimate of the sum of multiple sine series with linear phase. Next, as Theorem 2, a result established by the author and \textit{G. I. Arkhipov} [Mat. Sb., N. Ser. 134(176), No. 2(10), 147--157 (1987; Zbl 0665.42003)], which provides an estimate for the one-dimensional discrete Hilbert transform with polynomial phase. The main result of this paper is given in Theorem 3, which is on the multiple discrete Hilbert transform with polynomial phases. It generalizes and sharpens Theorems 1 and 2. To facilitate the proof of the main theorem five lemmas are established.